Lemma 15.89.5 (05ED)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.89: Formal glueing of module categories
Lemma 15.89.5 (cite)

Lemma 15.89.5. Let $R$ be a ring. Let $I = (f_1, \ldots , f_ n)$ be a finitely generated ideal of $R$ . Let $M$ be the $R$ -module generated by elements $e_1, \ldots , e_ n$ subject to the relations $f_ i e_ j - f_ j e_ i = 0$ . There exists a short exact sequence

$0 \to K \to M \to I \to 0$

such that $K$ is annihilated by $I$ .

Proof. This is just a truncation of the Koszul complex. The map $M \to I$ is determined by the rule $e_ i \mapsto f_ i$ . If $m = \sum a_ i e_ i$ is in the kernel of $M \to I$ , i.e., $\sum a_ i f_ i = 0$ , then $f_ j m = \sum f_ j a_ i e_ i = (\sum f_ i a_ i) e_ j = 0$ . $\square$

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